# How do you factor 3a^2 + 14a − 24?

Apr 6, 2016

$\left(3 a - 4\right) \left(a + 6\right)$

#### Explanation:

We know there must be a $3 a$ and $a$ in the brackets because they are the only factors of $3$.

You then use trial and error to find a the other numbers which multiply together to make $24$ and add together (multiplied by their opposite coefficients of $a$) to make $14$, which you find are $6$ and $- 4$.

Apr 6, 2016

(3a - 4)(a + 6)

#### Explanation:

Use the systematic, non-guessing new AC Method (Socratic Search).
$y = 3 {a}^{2} + 14 a - 24 =$ 3(a + p)(a + q)
Converted trinomial $y ' = {a}^{2} + 14 a - 72 =$ (a + p')(a + q')
p' and q' have opposite signs because ac < 0.
Factor pairs of (ac = -72) --> (-3,24) (-4, 18). This sum is 14 = b. Then,
p' = -4 and q' = 18.
Back to trinomial y, $p = \frac{p '}{a} = - \frac{4}{3}$ and $q = \frac{q '}{a} = \frac{18}{3} = 6$
Factored form: $y = 3 \left(a - \frac{4}{3}\right) \left(a + 6\right) = \left(3 a - 4\right) \left(a + 6\right)$